Map-Making and Power Spectrum Estimation for Cosmic Microwave Background Temperature Anisotropies

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HELSINKI INSTITUTE OF PHYSICS INTERNAL REPORT SERIES HIP-2005-04

Map-Making and Power Spectrum Estimation for Cosmic Microwave Background

Temperature Anisotropies

Torsti Poutanen

Helsinki Institute of Physics, and

Division of Theoretical Physics, Department of Physical Sciences Faculty of Science

University of Helsinki

P.O. Box 64, FIN-00014 University of Helsinki Finland

ACADEMIC DISSERTATION

To be presented for public criticism, with the permission of the Faculty of Science of the University of Helsinki, in Auditorium D101 at Physicum, Gustaf H¨ alltr¨ omin katu 2,

on November 24, 2005, at 2 p.m..

Helsinki 2005

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ISBN 952-10-1693-0 (printed version) ISSN 1455-0563

Helsinki 2005 Yliopistopaino

ISBN 952-10-1694-9 (pdf version) http://ethesis.helsinki.fi

Helsinki 2005

Helsingin yliopiston verkkojulkaisut

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Acknowledgements

This thesis is based on research carried out at Helsinki Institute of Physics and at the Theoretical Physics Division of the Department of Physical Sciences at the University of Helsinki during the years 2002-2005. I wish to thank the personnel of both institutes for a pleasant working atmosphere. I am thankful to the Väisälä Foundation and Academy of Finland for their financial support, and CSC - Scientific Computing Ltd. for computational resources.

I thank Prof. Kari Enqvist for his open-mindedness to accept me, who had no previous background in cosmology, to work in his Cosmology Group. I acknowledge his support during the entire time that I have pursued my thesis in his group.

My warmest thanks go to my supervisor and colleague Dr. Hannu Kurki-Suonio.

I greatly appreciate his deep interest to my work and the countless hours that he has spent discussing CMB data processing with me. I have felt priviledged to have such a devoted supervisor. His lecture notes on CMB physics were a great help when I wrote Chapter 2 of this thesis.

Dr. Elina Keihänen has been my closest coworker in the practical development of the CMB data processing methods. I thank her for many good advice and numerous tips in Fortran. I thank Ville Heikkilä for his help in CMB simulations and Vesa Muhonen and Jussi Väliviita for discussions on various CMB topics.

I am grateful to Davide Maino for his help in destriping and cooperation in papers I - III. I thank Carlo Burigana for cooperation in paper I.

I thank the referees of this thesis, Dr. Anthony Challinor and Dr. Ben Wandelt, for the careful reading of the manuscript.

Part of my thesis has benefitted from the collaboration of the Planck Working Group 3 (WG3). I thank Mark Ashdown, Carlo Baccigalupi, Amedeo Balbi, Julian Borrill, Chris Cantalupo, Giancarlo de Gasperis, Kris G´orski, Eric Hivon, Charles Lawrence, Paolo Natoli, Martin Reinecke, and Radek Stompor for cooperation in WG3. I acknowledge National Energy Research Scientific Computing Center of U.S Department of Energy for providing computational resources to WG3.

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This work has made use of the Planck satellite simulation package (Level S), which is assembled by the Max Planck Institute for AstrophysicsPlanck Analysis Centre (MPAC). I acknowledge the use of the CMBFAST code for the computation of the theoretical CMB angular power spectra. Some of the results in this thesis have been derived using the HEALPix package ([51],[52]).

Helsinki 2005 Torsti Poutanen

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Abstract

The universe is filled with cosmic microwave background (CMB) radiation. We can detect these photons in every direction with nearly the same intensity. The tiny intensity variation is called the CMB temperature anisotropy and it reflects the inhomogeneities of the universe during the photon decoupling. Cosmological theories, that describe the early universe, predict the statistical properties of these anisotropies. Therefore CMB observations are important when determining the values of the cosmological parameters of these theories. In this thesis we restrict ourselves to temperature observations and do not consider polarization here.

The CMB temperature anisotropy can be displayed as a pixelized map over the celestial sphere. Map-making and angular power spectrum estimation are important steps in the data processing of a CMB experiment. Destriping is an efficient method to reduce the level of correlated (1/f) noise in the observed data. In this thesis we have developed a maximum-likelihood approach to destriping and use these methods to make maps from the simulated observations of the Low Frequency Instrument (LFI) of the Planck satellite. We compare these output maps to the output maps of general least squares (GLS) map-making algorithms. Under the assumption of Gaussian distributed noise GLS algorithms are implementations of the maximum- likelihood map-making. Therefore their output maps fall close to the minimum variance map. Destriped maps are not optimal in this maximum-likelihood sense. Our results reveal, however, that the difference in the map noise between destriped and GLS output maps is very small in these cases. The map-making methods cause error in the signal part of the output maps too. The source of this error is the subpixel structure of the signal. Its coupling to the output map varies in different map-making methods. This error was larger for GLS than for destriping, but in both cases it was clearly smaller than the level of the noise in the maps.

In this thesis we also studied the angular power spectrum estimation from the output maps of destriping. We found that the map-making error due to pixelization noise had an insignificant effect in the power spectrum estimates. We noticed, however, that the non-uniform distribution of observations in the output map pixels caused high-` excess power in the power spectrum estimates. We corrected for this by subtracting a signal bias whose value we estimated using Monte Carlo simulations. The angular power spectrum estimates, that we obtained, were unbiased and their errors were close to their theoretical expectations.

Poutanen, Torsti: Map-Making and Power Spectrum Estimation for Cosmic Microwave Background Temperature Anisotropies, University of Helsinki, 2005, 144 p., Helsinki Institute of Physics In- ternal Report Series, HIP-2005-04, ISSN 1455-0563, ISBN 952-10-1693-0 (printed version), ISBN 952-10-1694-9 (pdf version).

Classification (INSPEC): A9575P, A9870V, A9880L

Keywords: cosmology, cosmic microwave background radiation, data analysis

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List of Papers

This thesis consists of three research papers and an introduction, which provides relevant background and additional discussions.

I E. Keih¨anen, H. Kurki-Suonio, T. Poutanen, D. Maino, and C. Burigana A MAXIMUM LIKELIHOOD APPROACH TO THE DESTRIPING TECH- NIQUE

A&A 428, 287-298 (2004). Reference [1] in the bibliography.

II T. Poutanen, D. Maino, H. Kurki-Suonio, E. Keih¨anen, and E. Hivon

COSMIC MICROWAVE BACKGROUND POWER SPECTRUM ESTIMA- TION WITH THE DESTRIPING TECHNIQUE

MNRAS 353, 43-58 (2004). Reference [2] in the bibliography.

III T. Poutanen, G. de Gasperis, E. Hivon, H. Kurki-Suonio, A. Balbi, J. Borrill, C. Cantalupo, O. Dor´e, E. Keih¨anen, C. Lawrence, D. Maino, P. Natoli, S.

Prunet, R. Stompor, and R. Teyssier

COMPARISON OF MAP-MAKING ALGORITHMS FOR CMB EXPERI- MENTS

To be published in A&A, [astro-ph/0501504] (2005). Reference [3] in the bibliography.

The contribution of the author to the publications

Paper I: This is the first paper that Keih¨anen, Kurki-Suonio and I published on CMB data processing. Although the destriping codes used in our paper were not written by me, I had made my own test codes using similar principles. That expe- rience allowed me to contribute in the development of the destriping theory and in the analysis of the results presented in the paper. I made inputs to the maximum- likelihood and to the eigenvalue analysis of the destriping problem.

Paper II: In this paper we studied the angular power spectrum estimation from the CMB temperature maps that were made using destriping. The items affecting

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the accuracy of the power spectrum estimates were analyzed using Monte Carlo simulations. I implemented the Monte Carlo pipeline, ran all the simulations and calculated the results shown in the paper. The destriping code used in the study was a code described in paper [1]. I wrote most of the the paper and developed the theory in its Appendix A.

Paper III: This paper is a result of the Planck WG3 collaboration. The maps for this study were made by research groups participating in WG3. I made the destriped maps using the code described in paper [1]. The methods used in the map and angular power spectrum comparison were mainly designed by me and I produced all the results. I wrote large parts of the paper and I developed a significant part of the theory in its Appendices A and B.

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List of Figures

2.1 Acoustic oscillations of the photon brightness function . . . 26

2.2 C` and cosmological parameters - 1 . . . 28

2.3 C_` and cosmological parameters - 2 . . . 29

3.1 Schematic view of the Planck satellite . . . 32

3.2 Detailed view of the Planck focal plane . . . 33

3.3 Foot-print of the focal plane in the sky . . . 34

3.4 Front-end unit of the LFI radiometer . . . 36

3.5 Back-end unit of the LFI radiometer . . . 36

3.6 1/f noise reduction in the LFI radiometer . . . 37

3.7 HFI focal plane unit . . . 38

3.8 Elliptic Gaussian beam pointing to the north pole . . . 40

3.9 Spherical harmonic expansion coefficients of an elliptic Gaussian beam pointing to the north pole . . . 41

3.10 Radiometer noise PSD . . . 45

3.11 Radiometer noise stream . . . 47

3.12 Planck in L2 orbit . . . 49

3.13 Coverage maps of two scanning strategies . . . 50

3.14 Map of the CMB dipole and galaxy . . . 52

3.15 TOD of the CMB dipole and galaxy . . . 53

3.16 Intensities of the galactic and CMB emissions . . . 54

3.17 Observed TOD . . . 56

3.18 Data processing pipeline . . . 58

4.1 Noise filters . . . 64

4.2 Uniform baselines modelling the 1/f noise . . . 67

4.3 Noise maps before and after destriping . . . 72

4.4 Angular power spectra of the instrument noise maps . . . 73

4.5 Observed temperature map after destriping . . . 80

4.6 Map-making error due to pixelization noise . . . 84

5.1 Hit count maps in the power spectrum study . . . 100

5.2 Angular power spectra of the sky coverage maps . . . 102

5.3 Mode coupling matrix . . . 103

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5.4 Spectrum of the binned noiseless map . . . 108

5.5 Ratio of power spectra with signal bias removed . . . 109

5.6 Noise bias . . . 111

5.7 Normalized estimator variance for different pixel weights . . . 115

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List of Tables

2.1 Cosmological parameters . . . 27

3.1 Antenna temperature of the CMB photons . . . 43

3.2 Goals of the Planck payload . . . 48

3.3 Amount of data produced by the Planck detectors . . . 55

3.4 Angular sizes of the HEALPix pixels . . . 59

4.1 Simulation parameters of the map comparison . . . 81

4.2 Results of the comparison of the temperature maps . . . 82

4.3 Signal component of the reconstruction error map . . . 83

5.1 Simulation parameters of the power spectrum estimation . . . 99

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Chapter 1 Introduction

The universe is filled with cosmic microwave background (CMB) radiation. We can detect these photons in every direction with nearly the same intensity. They have a blackbody spectrum of mean temperature T0 = 2.725 K ([4]). The CMB radiation was first observed in 1965 ([5]). There is a small temperature difference δT between photons coming from different directions. This is called the CMB temperature anisotropy. Its typical magnitude is δT /T0 .10⁻⁴. The CMB anisotropy was first observed by COBE satellite in 1992 ([6]).

The CMB photons were generated during photon decoupling, where photons became free from the baryon interactions due to the formation of stable atoms. Photon decoupling took place about 380 000 years after the big bang ([7]). The CMB temperature anisotropy, that we see today, reflects the inhomogeneities of the universe during the photon decoupling and, to a small extent, also later inhomogeneities.

Theories, that describe the early universe, predict the statistical properties of the CMB anisotropies that we observe today. Therefore CMB observations are important when we choose between these theories and determine the values of the cosmological parameters.

The CMB temperature anisotropy is usually displayed in a pixelized map over the celestial sphere. The angular power spectrum of that map is a quantity whose value is predicted by the cosmological theories. Map-making and angular power spectrum estimation from the observed data are important steps in the data processing of a CMB experiment. In this thesis we restrict ourselves to the map-making and power spectrum estimation from the temperature anisotropy observations. We do not consider polarization here.

We introduce the reader to the CMB data processing from two directions. In Chapter 2 we give a brief discussion on CMB physics and show how the angular power spectrum, that we observe today, depends on the inhomogeneities of the universe during photon decoupling. In Chapter 3 we describe the relevant aspects of a satellite CMB experiment that we need to know in order to understand the observed data that we use as an input in the map-making. We use the Planck experiment as an example here ([8], [9]).

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The noise of the detectors is an important source of error in a CMB experiment.

In this thesis we assume that the detector noise is a sum of correlated (1/f) and non-correlated (white) noise. The correlated part of the noise, when coupled to the observing strategy, will lead to stripes in the final maps. Destriping is a map- making method which, in its simplest form, models the correlated noise with uniform baselines and uses the observations of pixels, that are monitored several times during the mission, to solve the estimates of the baseline magnitudes. When the baselines have been subtracted from the observed data, the cleaned data can be averaged (binned) to the output map pixels.

Destriping was first introduced to the CMB map-making in the early planning phases of the Planck experiment ([68]). It is an efficient map-making algorithm which requires no prior knowledge on the instrument noise. However, it is not optimal in the sense that it does not produce a minimum variance map. Under the assumption of Gaussian distributed noise the maps produced by the general least squares (GLS) map-making algorithms ([56] - [61]) produce maps that fall closer to the minimum variance map.

In this thesis we developed a maximum likelihood approach to destriping and implemented map-making codes that use these principles (paperI[1]). Compared to the destriping methods of other authors ([69] - [73]) our method differs in the weights it gives to the observed pixels. Our weights arise from the maximum-likelihood analysis while the weights of the other authors are more heuristic. We compare in paperI([1]) the maps of our weights to the maps of the weights of the other authors.

Using simulated observations of a Planck detector we also compared the output maps of our implementation of the destriping method and two implementations of the GLS method (paper III[3]).

A discussion of the map-making problem and map-making algorithms is given in Chapter 4 of this thesis. The results of the above map comparisons are given there as well.

In this thesis we also studied angular power spectrum estimation from the output maps of destriping (paper II [2]). We used Monte Carlo methods to reveal the accuracy of our power spectrum estimates. A thorough discussion of the angular power spectrum estimation and the results of the study (paper II [2]) are given in Chapter 5 of this thesis. Appendices A and B provide some further details to Chapter 5.

Chapters 2 and 3 are introductions to the map-making and angular power spectrum estimation, which are the main research topics of this thesis. They are discussed in Chapters 4 and 5 and in papersI - IIIof this thesis. We did not carry out any research on CMB physics in this thesis. Therefore Chapter 2 just gives a brief overview of this topic.

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Chapter 2 CMB Physics

2.1 Introduction

When the universe had cooled to T ≈ 3000 K, protons and electrons were able to form stable hydrogen atoms. Before that all matter was in a plasma state and the photon mean free path was short due to Compton scattering from the charged particles and ions (atomic nuclei). Frequent scattering maintained the photons and baryons in a thermal equilibrium and they can be described as a single ”baryon- photon” fluid at that time. The build-up of hydrogen atoms decreased the density of free electrons and the mean free path of the photons was rapidly increased allowing them to propagate nearly freely. The epoch when the photons became free from the baryon-photon fluid is called the photon decoupling. It occurred about 380 000 years after the big bang at redshift z_dec ≈ 1089 ([7]). It was not an instantaneous event but it lasted about 120 000 years ([7]). The value of a quantity at the photon decoupling is denoted with a subscript ”dec” in this chapter.

Photons liberated at decoupling have propagated to us and we can detect them today. We can see them coming to us from all directions with a nearly constant intensity. The stream of these photons is the CMB radiation. Before the CMB was detected by Penzias and Wilson in 1965 ([5]) its existence was theoretically predicted (see e.g. [10] - [14]). The mean temperature of the CMB photons, that we see today, has redshifted to T0 = 2.725 K ([4]).

Already before the detection of the CMB it was anticipated that the energy density (ρ) of the early universe (before photon decoupling) was not perfectly homogeneous but it had some small non-homogeneous perturbations (see e.g. [15] - [17])

ρ(t,x) = ¯ρ(t) +δρ(t,x). (2.1) Here ¯ρ(t) is the homogeneous part of the energy density. It has the same value in every space-point of the universe and it depends only on time (t). The non- homogeneous energy-density perturbationδρ(t,x) depends both on space (indicated by 3-vector x) and time. It is assumed that the energy density perturbations were

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small (|δρ(t,x)| ¿ ρ(t)) in the early universe. The regions of energy overdensity of¯ the early universe have grown to the galaxies that we see today.

The energy density perturbations caused perturbations in the intensity of CMB photons. These CMB perturbations were predicted theoretically (see e.g. [15] - [17]) and they were detected by the COBE satellite in 1992 ([6]). According to the COBE observations the CMB photons coming to us from different directions have slightly different temperatures. The relative differences are 10⁻⁵. . .10⁻⁴. This variation (relative to the mean temperatureT0) is called the CMB anisotropy signal. It has a distinct value in every point of the celestial sphere and the observations of the CMB anisotropy field are usually displayed as a map over the sphere.

It is widely believed that the universe underwent a period of an exponential expansion in its early history (. 10⁻³² s after the big bang). This expansion is called inflation. One or more scalar fields (inflaton fields) could have caused the rapid expansion. Vacuum fluctuations of those fields lead to perturbations in the energy-momentum tensor. The inflation model predicts that the energy density perturbations were initiated during the inflation period (∼10⁻³² s after the big bang).

In this chapter of this thesis we apply units where c=~=kB = 1.

2.2 Perturbed Universe

Because we assume that the early universe was nearly homogeneous with some small non-homogeneous perturbations, it can be described with a metricg_µν that is a sum of a metric of a homogeneous spacetime and a small non-homogeneous perturbation

gµν = ¯gµν+δgµν. (2.2)

The homogeneous universe with the metric ¯g_µν is called the background universe.

The metric of the perturbed universe is gµν(t,x).

Because we assume that the perturbations are small (relative to their background values), our perturbed universe can be approximately described by first order perturbation theory of general relativity. In this theory every quantity is a sum of its background quantity and a small perturbation: h = ¯h +δh. We denote with an overbar the quantities of the background universe. They depend on time only. We will drop from all equations the terms which are of order O(δh²) or higher. Thus the perturbation of a quantity depends on the perturbations of the other quantities through linear equations.

In general relativity the development of the universe is determined by the Ein- stein equations

G^µ_ν = 8πGT_ν^µ. (2.3)

HereT_ν^µ is the energy-momentum tensor, G is the gravitational constant and G^µ_ν is the Einstein tensor. The latter depends on the metric tensor and its first and second derivatives with respect to the spacetime coordinatesx^µ. The Einstein equations are valid separately in the perturbed and in the background universes. Subtracting the

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Einstein equations of these universes from each other leads to the Einstein equations between the perturbations δG^µ_ν and δT_ν^µ.

The fact thatG^µ_ν;µ= 0 (subscript ”;µ” means a covariant derivation with respect tox^µ) leads to energy-momentum continuity equationsT_ν;µ^µ = 0. They apply in both perturbed and background universes. Energy-momentum continuity equations can be used instead of some of the Einstein equations.

In this chapter we assume, for simplicity, that the background universe is a flat Friedmann-Robertson-Walker (FRW) universe. Its metric in comoving coordinates (t, x, y, z) is

ds² = ¯gµνdx^µdx^ν =−dt²+a²(t)[dx²+dy²+dz²]. (2.4) In this chapter we apply a summation rule where we sum over the repeated indeces (Einstein summation rule). The function a(t) is the scale factor.

We will often use the conformal time η instead of the coordinate time t. They are related by dη =dt/a(t). Using conformal time the background metric is

ds² =a²(η)[−dη²+dx²+dy²+dz²]. (2.5) We assume that the background energy-momentum tensor can be approximated by the perfect fluid energy-momentum tensor

T_ν^µ=diag(−ρ,¯ p,¯ p,¯ p),¯ (2.6) where ¯p is the background pressure of the fluid. The equations for the scale factor can be obtained from the 0−0 and i−i components of the background Einstein equations. These equations (for the scale factor) are called the Friedmann equations.

We give them here using the conformal time.

H² ≡ µa⁰

a

¶2

= 8πG

3 ρa¯ ² (2.7)

and

H⁰ =−4πG

3 (¯ρ+ 3¯p)a², (2.8)

where⁰ ≡d/dη and H ≡a⁰/a is the comoving Hubble parameter. Its relation to the ordinary Hubble parameterH isH=aH. Unless otherwise noted we normalize the scale factor to have value a0 = 1 today (subscript ”0” of a quantity means its value today). In this normalization a comoving value of a quantity equals its value today.

The continuity equation ¯T_0;µ^µ = 0 of the background universe gives

¯

ρ⁰ =−3H(¯ρ+ ¯p). (2.9)

A thorough discussion of the perturbation theory in the FRW universe is given in e.g. [18] - [20]. We just give some main results here.

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The perturbations in the metric and energy-momentum tensors can be scalar, vector or tensor type depending on how they behave in the 3-space coordinate rota- tions. In linear perturbation theory these three types of perturbations evolve independent from each other. The scalar perturbations are the most important, because they couple to the energy density and pressure perturbations and are responsible for the gravitational growth of the overdense regions of the early universe. We will consider only the scalar perturbations in this chapter and ignore the vector and tensor perturbations.

In the scalar perturbation theory the perturbation B_i of a 3-vector quantity (i indexes the space coordinates) is derived from a scalar perturbation B: Bi = −B,i

(”, i” is a partial derivative with respect to xⁱ). The perturbation Eij of a 3-space traceless tensor quantity is derived from a scalar perturbation as well: E_ij =E_,ij −

1

3δij∇²E. Here δij is the Kronecker delta.

There are several coordinate systems in the perturbed universe that are close to each other and that we could therefore use. Transformations between such coordinate systems are called gauge transformations. For practical calculations we need to select a coordinate system. In this thesis we use the conformal-Newtonian gauge. In this gauge the metric is

ds² =a²(η)[−(1 + 2Φ)dη²+ (1−2Ψ)(dx²+dy²+dz²)]. (2.10) Here Φ and Ψ are small perturbations from the background metric. They are functions of the spacetime coordinates (η,x) and are called Bardeen potentials ([19]).

The energy-momentum tensor of the perturbed universe is (scalar perturbations in the conformal-Newtonian gauge)

T_ν^µ=

· −ρ¯

¯ pδij

¸ +

· −δρ −(¯ρ+ ¯p)v,i

(¯ρ+ ¯p)v,i δpδij + ¯pΠ,ij −¹₃pδ¯ ij∇²Π

¸

(2.11) Here index µ refers to the rows and index ν refers to the columns of the energy- momentum tensor. The quantities δρ and δp are the energy density and pressure perturbations and−v,i is the velocity perturbation (derived from the scalar perturbation v) of the fluid. The background value of the velocity perturbation is zero.

The quantity Σij = ¯pΠ,ij−¹3pδ¯ ij∇²Π is the anisotropic stress. Its background value is zero as well.

Because we assume a flat background universe, the perturbations can be Fourier expanded using the plane waves e^ik·x (complete orthogonal set in a flat universe).

We can write for the energy density perturbations ([20]) δρ(η,x) = 1

(2π)^3/2 Z

d³kδρ(η,k)e^ik·x. (2.12) Herekis the comoving wavevector. Its magnitudek (k ≡ |k|) can be given in terms of the comoving wavelengthλ:k = 2π/λ. The physical wavelength isλphys =aλ and the physical wavevector is kphys =ka⁻¹. The physical wavelength of a Fourier mode

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kincreases as the universe expands. The comoving wavelength of a Fourier mode k is called a scale. The unit vector in the direction of k is ˆk (k = kk) and ˆˆ kⁱ is the component of ˆk in the xⁱ direction. Similar Fourier expansions exist for δp, Ψ and Φ.

Following [20] we define for the Fourier expansions of the scalar velocity perturbation v and the scalar anisotropic stress Π

v(η,x) = 1 (2π)^3/2

Z

d³kv(η,k)

k e^ik·x (2.13)

and

Π(η,x) = 1 (2π)^3/2

Z

d³kΠ(η,k)

k² e^ik·x. (2.14)

This way v(η,k) and Π(η,k) will have the same dimensions and magnitudes as the perturbations (vi = −v,i and Σij/¯p) themselves have. In first order perturbation theory each Fourier mode evolves independently. We can find a solution to each Fourier mode separately and obtain the total perturbations from Eqs. (2.12) - (2.14).

We can now apply the Fourier expansion to the Einstein equations and we obtain for the Fourier modes of the perturbations

H⁻¹Ψ⁰ + Φ +1 3

µk H

¶2

Ψ = −1

2δ (2.15)

H⁻¹Ψ⁰+ Φ =−3

2(1 +w)H

k v (2.16)

H⁻²Ψ⁰⁰+H⁻¹(Φ⁰+ 2Ψ⁰)−3wΦ−1 3

µk H

¶2

(Φ−Ψ) = 3 2

δp

¯

ρ (2.17)

µk H

¶2

(Ψ−Φ) = 3wΠ (2.18)

We have defined the density contrastδ ≡δρ/¯ρ and the equation-of-state parameter w ≡p/¯ ρ. For later convenience we also define the ”speed of sound”¯ cs of the fluid:

c²_s ≡p¯⁰/¯ρ⁰. Note that all perturbations are functions of (η,k).

We can Fourier expand the continuity equations as well and obtain the following evolution equations for the Fourier modes of the perturbations

δ⁰ = (1 +w)(−v+ 3Φ⁰) + 3H µ

wδ− δp

¯ ρ

¶

(2.19) v⁰ =−H(1−3w)v− w⁰

1 +wv+ δp

¯

ρ+ ¯p −2 3

w

1 +wΠ + Φ (2.20) The fluctuations in the local curvature of the perturbed universe are charac- terised by the curvature perturbation R(η,x). In the conformal-Newtonian gauge it is defined as

R=−Ψ− 2

3(1 +w)(H⁻¹Ψ⁰+ Φ). (2.21)

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It gives the local curvature of a t = const slice in the comoving gauge, i.e. one where the fluid flow is orthogonal to this slice. Although we do not work here in the comoving gauge, R is a useful quantity to describe primordial perturbations, since, as explained below, for adiabatic perturbations it remains constant at scales outside horizon (”outside horizon” means here that λÀ H⁻¹ which is equivalent to k ¿ H).

Using Einstein equations an evolution equation can be derived for the Fourier modes of the curvature perturbation

3

2(1 +w)H⁻¹R⁰ = µk

H

¶2·

c²_sΨ + 1

3(Ψ−Φ)

¸ +9

2c²_s(1 +w)H µδp

¯ p⁰ − δρ

¯ ρ⁰

¶

. (2.22) In this thesis we assume that the perturbations are initially adiabatic. Adiabatic perturbations are predicted by the simple inflation models and are consistent with observations. For adiabatic perturbations δp/p¯⁰ =δρ/ρ¯⁰ and they remain adiabatic while outside horizon. Thus the second term on the right hand side of Eq. (2.22) is zero.

At the end of the inflation all relevant scales were outside horizon. For these scales the first term on the right hand side of Eq. (2.22) is small. Therefore, for the adiabatic curvature perturbations, whose scales are outside horizon, R⁰ = 0. This means that the curvature perturbations of those scales remain constant (in time) as long as they are outside horizon.

To solve the perturbations of the interesting quantities at some time (e.g. today) we need to set the initial conditions. They are usually specified during the early radiation-dominated era (t = trad), when all interesting scales were well outside the horizon (k ¿ H). We can assume that at this time neutrino decoupling, e⁺e⁻ annihilation and nucleosynthesis were over and the temperature of the universe was, e.g., T≈10⁷ K. Because all relevant scales are outside horizon, the Fourier modes of the curvature perturbation have constant values which we denote by R^k(rad). They are called the primordial values of the curvature perturbation. They are our initial conditions. The magnitude of the first order perturbation of a quantity at some time later than trad depends linearly onR^k(rad). Because att > trad the horizon expands more rapidly than the perturbation scales, the scales enter the horizon one by one (smaller scales enter earlier than the larger ones) and theirR evolve thereafter.

Present theories, like inflation, for the origin of the perturbations assume that their primordial values have been produced by random processes. Therefore these theories do not give us the primordial values themselves but their statistical properties. It is usually assumed that R^k(rad) are zero mean, complex Gaussian random variables with a power spectrum ([20])

hR^k(rad)R^∗k⁰(rad)i= 2π²

k³ PR(k)δ(k−k⁰). (2.23) Here asterisk indicates a complex conjugate andh·iis an ensemble average. Typical inflation theories predict thatP^R(k)∝kⁿ⁻¹, wheren≈1 (scale invariant spectrum).

The parameter n is called the spectral index.

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2.3 Boltzmann Equation for the CMB Photons

Using Eqs. (2.15) - (2.20) and the initial conditions, the evolution of the fluid perturbations can be determined. These equations govern the fluid as a whole but they are not able to tell us anything about the evolution of different particle species of the fluid. At t > t_rad we assume that the universe contains photons, baryons (electrons, protons, neutrons, ions, atoms), cold dark matter (CDM) and neutrinos. Although electrons are not really baryons (they are leptons), they are tightly coupled to baryons, forming a single fluid component with them, and therefore it is customary in cosmology to include electrons under the term ”baryons”. Photons and baryons interact with Compton scattering, charged baryons are coupled with an electromagnetic interaction, neutrons and protons with the strong interaction, and all particles contribute to the gravity and are affected by it.

To account for the particle interactions we need to discuss the distribution functions of each particle species and examine their Boltzmann equations that govern their evolution. Boltzmann equations of different particle species are discussed in e.g.

[20] - [24]. The issues discussed in the remaining parts of this chapter are extracted from these references. This section covers the Boltzmann equation for the CMB photons. We are restricted to the CMB temperature anisotropies. The Boltzmann equations for polarization are discussed in e.g. [25].

2.3.1 Photon Distribution and Brightness Functions

The photon distribution function f(t,x,p) is defined in 6-dimensional phase space so that _(2π)^g 3f is the number of photons in a phase space element d³xd³p. Here g

= 2 is the number of spin states, (t,x) are the comoving coordinates and p is the photon momentum in the locally orthonormal coordinates of the comoving observer (i.e., one who is at rest in the (t,x) coordinate system). The distribution function evolves in time due to freely falling motion of the photons and due to collisions with charged baryons. The evolution is governed by the Boltzmann equation

df

dt =C[f], (2.24)

where C[f] is the collision term. The collisionless Boltzmann equation (df /dt = 0) is discussed in Sect. 2.3.2 and the collision term is given in Sect. 2.3.3 of this thesis.

In first order perturbation theory the distribution function is a sum of the background function and a perturbation. The background distribution function can be approximated by the distribution function of the thermal equilibrium (blackbody distribution function)

f¯(t, p) = 1

e^p/T^(t)−1. (2.25)

Herep=|p|, which equals the photon energy (in the locally orthonormal coordinate system). Today the value of the temperature T(t) is T(t₀) = T₀ = 2.725 K. The

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perturbed distribution function can be given in the following form f(t,x, p,p) =ˆ 1

exp³

p T(t)[1+Θ(t,x,ˆp)]

´−1. (2.26)

Here p = pˆp and ˆp is the unit vector in the direction of the photon momentum.

The function Θ(t,x,p) is called theˆ brightness function. It does not depend on the full momentum, just on its direction. This reflects the usual approximation (justified in Sect. 2.3.3), where the perturbation depends on the direction of the photon momentum, but it is not a deviation from the blackbody spectrum ([24]).

The perturbed distribution function can be expanded (using Taylor series) in the sum of the background function and a perturbation

f(t,x, p,p) = ¯ˆ f(t, p) +δf(t,x,p) = ¯ˆ f+ ∂f¯

∂TTΘ = ¯f−p∂f¯

∂pΘ. (2.27) The last form is obtained after applying the identity T ∂f /∂T¯ =−p∂f /∂p.¯

2.3.2 Collisionless Boltzmann Equation

The derivative of the photon distribution function can be written as df

dt = ∂f

∂t + ∂f

∂xⁱ dxⁱ

dt +∂f

∂p dp

dt + ”∂f

∂pˆ dˆp”

dt (2.28)

The background distribution function does not depend on the photon direction and the photons do not change direction in the background universe. Therefore the terms

∂f /∂pˆ and dˆp/dt are perturbations and their product can be ignored (in first order perturbation theory).

The relation between the photon 4-momentum (P⁰, Pⁱ) in the coordinate basis and its momentum (p, pⁱ) in the locally orthonormal frame of the comoving observer is (no summation over repeated indeces)

p=p

|g00|P⁰ (2.29)

and

pⁱ =p

|gii|Pⁱ (2.30)

Usingpⁱ =pˆpⁱ (ˆpⁱ is thei^th component of the unit vector ˆpof the photon momentum and δ_ijpˆⁱpˆ^j = 1) and the conformal-Newtonian metric (Eq. (2.10)), we obtain

P⁰ =a⁻¹(1−Φ)p (2.31)

and

Pⁱ =a⁻¹(1 + Ψ)ppˆⁱ. (2.32)

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For the second term of Eq. (2.28) we need to evaluate dxⁱ/dt.

dxⁱ dt = 1

a dxⁱ

dη = 1 a

dxⁱ dλ

dλ dη = 1

a Pⁱ P⁰ = 1

a 1 + Ψ

1−Φpˆⁱ =a⁻¹(1 + Φ + Ψ)ˆpⁱ. (2.33) Hereλ is the affine parameter of the photon trajectory (P^µ =dx^µ/dλ) and we used Eqs. (2.31) and (2.32) in place of P⁰ and Pⁱ. Because the background distribution function does not depend onxⁱ, the term ∂f /∂xⁱ is a perturbation and we can thus drop Φ and Ψ from Eq. (2.33) when using it in the second term of Eq. (2.28). We obtain for the second term

∂f

∂xⁱ dxⁱ

dt = pˆⁱ a

∂f

∂xⁱ. (2.34)

The term dp/dt required in the third term of Eq. (2.28) can be evaluated using the photon geodesic equation

dP^µ

dλ + Γ^µ_αβP^αP^β = 0. (2.35)

Here Γ^µ_αβ are the Christoffel symbols of the metric (Eq. (2.10)). Dividing both sides of Eq. (2.35) with P⁰ =dη/dλ we obtain another form for the geodesic equation

dP^µ

dη + Γ^µ_αβP^αP^β

P⁰ = 0. (2.36)

Using the time component (µ = 0) of the geodesic equation and inserting Eqs.

(2.31) and (2.32) in place of P⁰ and Pⁱ, the derivative ^dp_dt = _adη^dp can be evaluated.

We will not show the details of this derivation here but merely give the final result.

The details of this calculation are given in e.g. [24].

dp dt =p

·

−H−pˆⁱ a

∂Φ

∂xⁱ +∂Ψ

∂t

¸

. (2.37)

We can now write the collisionless Boltzmann equation df

dt = ∂f

∂t + pˆⁱ a

∂f

∂xⁱ +p∂f

∂p

·

−H−pˆⁱ a

∂Φ

∂xⁱ +∂Ψ

∂t

¸

= 0. (2.38)

The first two terms on the right hand side are standard hydrodynamics. The first term in the brackets is the redshift due to expansion of the universe and the remaining terms represent the effects of the perturbations.

There is a collisionless Boltzmann equation in the background universe as well.

We can extract it from Eq. (2.38) df¯

dt = ∂f¯

∂t −Hp∂f¯

∂p = 0. (2.39)

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It is easy to show, that ¯f withT(t)∝1/ais a solution of the collisionless background Boltzmann equation. This is the redshift of the temperature of the background universe.

Subtracting Eq. (2.39) from Eq. (2.38) we obtain a collisionless Boltzmann equation for the brightness function ([24])

dδf

dt =−p∂f¯

∂p

·∂Θ

∂t + pˆⁱ a

∂Θ

∂xⁱ +pˆⁱ a

∂Φ

∂xⁱ − ∂Ψ

∂t

¸

= 0. (2.40)

The term inside the brackets must be zero. It can be expressed in terms of the conformal time

∂Θ

∂η + ˆpⁱ∂Θ

∂xⁱ + ˆpⁱ∂Φ

∂xⁱ −∂Ψ

∂η = 0. (2.41)

This is the collisionless brightness equation.

We can Fourier expand both sides of this equation and obtain the collisionless brightness equation for the Fourier modes of the photon brightness function

Θ⁰+ikµΘ +ikµΦ−Ψ⁰ = 0. (2.42) Hereµ≡ˆkⁱpˆⁱ = ˆk·pˆ is the cosine of the angle between the Fourier mode wavevector k and the photon momentum p and ⁰ ≡d/dη.

2.3.3 Collision Term

At the time of photon decoupling all baryons are moving with non-relativistic veloc- ities. Scattering of photons from non-relativistic charged baryons is called Thomson scattering. Its differential cross section is (e.g. [24])

dσ dΩ = 3

16πσT

¡1 + cos²(θ)¢

, (2.43)

where σT = ^8π₃ _m^α²2 is the total cross section of the Thomson scattering, α is the fine-structure constant and m is the rest mass of the baryon. The scattering angle between the directions of incoming and outgoing photon (in the baryon rest frame) isθ. Due to the 1/m² dependence Thomson scattering from the electrons is the only relevant scattering here.

Schematically (ignoring the stimulated emission and Pauli blocking) the collision term C[f] (see Eq. (2.24)) can be expressed as ([24])

C[f(p)] = X

q,q⁰,p⁰

|A|²[fe(q⁰)f(p⁰)−fe(q)f(p)]. (2.44) Here subscript ”e” refers to electron distribution function, qand q⁰ are the electron momenta before and after the collision, p and p⁰ are the corresponding photon momenta and |A|² is the scattering magnitude. It is directly proportional to the

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differential scattering cross section _dΩ^dσ. We are interested in the change of distribution of photons with momentum p. Therefore we sum over the other momenta.

The collision term C[f] is zero for the electron and photon background distribution functions ([24]). Thus the collision term is a perturbation and the collisional Boltzmann equation df /dt=C[f] splits in to df /dt¯ = 0 and dδf /dt=C[f].

The derivation of the collision term is a tedious task that we are not going to do here. We merely give the final result. The details of this calculation are given in e.g.

[24]. The collision term for photons is C[f(t,x,p)] = −p∂f¯

∂pneσT

·

Θ0−Θ + ˆp·vb+ 3

4pˆⁱpˆ^jΘ^ij₂

¸

. (2.45)

Here ne is the density of free electrons and vector vb is the velocity perturbation of the baryons (electrons are tightly coupled to the other baryons due to Coulomb interaction and they all have the same velocity perturbation). The velocity perturbation is a function of (t,x). The term ˆp·vb arises from the induced dipole in the rest frame of the scattering electron which is why it is independent of the photon brightness function Θ. The quantities Θ0 and Θ^ij₂ are the monopole and quadrupole of the photon brightness function. They are defined as

Θ0(t,x) = 1 4π

Z

dΩpˆΘ(t,x,p)ˆ (2.46) and

Θ^ij₂(t,x) = 1 4π

Z dΩpˆ

µ ˆ

pⁱpˆ^j − 1 3δ^ij

¶

Θ(t,x,p).ˆ (2.47) The quadrupole term arises from the angular dependence of Thomson scattering (see Eq. (2.43)). We ignore here the polarization dependence of Thomson scattering.

We finally obtain the collisional Boltzmann equation for the photons p∂f¯

∂p

·∂Θ

∂t + pˆⁱ a

∂Θ

∂xⁱ + pˆⁱ a

∂Φ

∂xⁱ − ∂Ψ

∂t

¸

=p∂f¯

∂pneσT

·

Θ0−Θ + ˆp·vb+3

¸ . (2.48) The only p dependent (photon energy dependent) term is p∂f /∂p. Because it can¯ be divided off from the both sides of Eq. (2.48), the brightness function remains independent from the photon energy. Therefore the perturbations do not change the frequency spectrum of the CMB photons and their spectrum today is still close to the blackbody spectrum.

We can equate the terms multiplying p∂f /∂p¯ in both sides of Eq. (2.49) and obtain the collisional brightness equation for photons

∂Θ

∂t + pˆⁱ a

∂Θ

∂xⁱ + pˆⁱ a

∂Φ

∂xⁱ − ∂Ψ

∂t =neσT

·

Θ0−Θ + ˆp·vb+3

¸

. (2.49) In terms of conformal time the collisional brightness equation is

∂Θ

∂η + ˆpⁱ∂Θ

∂xⁱ + ˆpⁱ∂Φ

∂xⁱ − ∂Ψ

∂η =an_eσ_T

·

Θ₀−Θ + ˆp·v_b+3

¸

. (2.50)

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We will obtain our final form of the collisional brightness equation after we have applied the following three steps in Eq. (2.50).

1. We have restricted ourselves to scalar perturbations in this chapter. Therefore we assume that the perturbationsvb and Θ^ij₂ are expressed in terms of scalar perturbations vb and Θ2: vb = −∇vb and Θ^ij₂ = (∂i∂j − ¹₃δij∇²)Θ2. Here ∂i

denotes partial derivation with respect to xⁱ.

2. We Fourier expand both sides of Eq. (2.50) using Eq. (2.12) (for Θ,Ψ,Φ), Eq.

(2.13) (forvb) and Eq. (2.14) (for Θ2). Thereafter we can extract the equation for the Fourier modes of the brightness function.

3. We define the optical depth τ(η) as an integral ofaneσT from conformal time η to the present time (η₀)

τ(η) = Z η0

η

dηaneσT. (2.51)

The significance of the optical depth is that the factor e^−τ represents the fraction of photons that have not been scattered between η and η0. Because aneσT =−τ⁰, we can use−τ⁰in place ofaneσT. For the later use we define here τr≡τ(ηreion), whereηreion is the time when the neutral gas (neutral since photon decoupling) between early stars and galaxies has become ionized because of the radiation from those stars and galaxies. This is called thereionization and according to WMAP observations the gas was reionized around zreion ≈ 20 ([7]), which corresponds to η = ηreion. The parameter τr is the optical depth due to reionization.

Now we are ready to write the collisional brightness equation for the Fourier modes of the photon brightness function (cf. Eq. (2.42) for the collisionless equation)

Θ⁰+ikµΘ +ikµΦ−Ψ⁰ =−τ⁰

·

Θ0−Θ−iµvb− 1

2P2(µ)Θ2

¸

. (2.52)

The function P2(µ) = ¹₂(3µ² −1) is the 2^nd order Legendre polynomial and the parameter µ= ˆk·pˆ was defined in Eq. (2.42). We remind the reader that here Θ is a function of (η,k,p) and Ψ, Φ,ˆ vb, Θ0 and Θ2 are functions of (η,k).

2.3.4 Boltzmann Hierarchy

Well before the photon decoupling photons and baryons were nearly in a thermal equilibrium. The collision term of the photon brightness equation (right hand side of Eq. (2.52) was small at that time. Because the number density of the free electrons (ne) was large (leading to large −τ⁰), the term inside the brackets was nearly zero leading to

Θ(η,k,p)ˆ ≈Θ0(η,k)−iµvb(η,k)−1

2P2(µ)Θ2(η,k) (2.53)

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We can see that before photon decoupling the brightness function depends on the direction of the photon momentum (ˆp) only via the parameter µ= ˆp·kˆ = cos(θ), where θ is the angle between ˆp and ˆk.

Assuming that ˆk is aligned with the z-axis of a cartesian coordinate system, θ is the elevation angle of the photon momentum in the corresponding spherical coordinate system (θ, ϕ). Because the collisional brightness equation involves only η, k and µ, the brightness function Θ remains a function of (η,k, µ) and no ϕ (azimuth) dependence will be developed. This allows us to expand Θ(η,k, µ) in terms of Legendre polynomials P_`(µ) (see e.g. [24])

Θ(η,k, µ) = X∞

`=0

(−i)^`(2`+ 1)Θ`(η,k)P`(µ). (2.54) The first three Legendre polynomials areP₀(µ) = 1,P₁(µ) =µand P₂(µ) = ¹₂(3µ²− 1). Using the orthogonality property

Z 1

−1

dµP`(µ)P`⁰(µ) = 2

2`+ 1δ``⁰ (2.55)

the above equation can be inverted to give the multipoles Θ`(η,k) of the photon brightness function

Θ`(η,k) =i^` Z 1

−1

dµ

2 P`(µ)Θ(η,k, µ). (2.56) A straightforward calculation shows that the functions Θ0 and Θ2 of Eq. (2.52) are indeed the monopole and quadrupole of Eq. (2.56).

We see from Eq. (2.53) that before the photon decoupling the brightness function contained essentially only monopole and dipole components (quadrupole is zero since it appears on both sides of Eq. (2.53) with different coefficients). The quadrupole and the higher multipoles develop after photon decoupling when the photons are travelling to us.

To obtain an equation for the multipoles we operate withi^`R1

−1dµP`(µ) on both sides of Eq. (2.52). Using the orthogonality of the Legendre polynomials (Eq. (2.55)) and the relation (2` + 1)µP`(µ) = (` + 1)P`+1(µ) + `P`−1(µ) we obtain for the multipoles of the photon brightness function

Θ⁰_`+ k

2`+ 1[(`+ 1)Θ`+1−`Θ`−1]− k

3Φδ`1−Ψ⁰δ`0 =

=−τ⁰

·

Θ0δ`0−Θ`+1

3vbδ`1+ 1 10Θ2δ`2

¸

. (2.57)

Using the photon distribution function the components of the photon energy- momentum tensor in the locally orthonormal frame can be calculated as

T_γ^µν = Z

d³pfp^µp^ν

p . (2.58)

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Here p =|p| is the energy of the photon in the locally orthonormal coordinates of the comoving observer and pⁱ are the space components of the photon momentum vector p.

We can equate this energy-momentum tensor to the energy-momentum tensor of Eq. (2.11) containing the photon perturbations. This gives the following relations between the photon perturbations and the multipoles of the brightness function

Θ0(η,k) = 1

4δγ(η,k), Θ1(η,k) = ¹₃vγ(η,k), Θ2(η,k) = 1

12Πγ(η,k). (2.59) Using these relations we can write for the multipoles

δ⁰_γ+4

3kvγ−4Ψ⁰ = 0 (` = 0) (2.60)

v_γ⁰ +k

6Πγ− k

4δγ−kΦ =τ⁰(vγ−vb) (`= 1) (2.61) Θ⁰₂+k

5[3Θ3−2Θ1] = 9

10τ⁰Θ2 (` = 2) (2.62) Θ⁰_`+ k

2`+ 1[(`+ 1)Θ`+1−`Θ`−1] =τ⁰Θ` (`≥3). (2.63) This system of coupled equations for the multipoles of the photon brightness function is called the Boltzmann hierarchy.

2.4 Boltzmann Equations for the Other Particle Species

We will not derive the Boltzmann equations for neutrinos, CDM and baryons in this chapter. We just give them here. The details of their derivations can be found in e.g. [23], [24].

Neutrinos were decoupled from the baryons well before the photon decoupling (neutrino decoupling at T ≈1 MeV). After their decoupling the massless neutrinos behave very much like photons, except that they are fermions and they interact very weakly with other particles. Their perturbations can be described with neutrino brightness function Θ^ν(η,x,p) that has a similar Boltzmann hierarchy as photons.ˆ The neutrino collision term is, however, zero.

The effect of neutrino mass is hardly detected in the present CMB data. There- fore we ignore the neutrino mass in this chapter. The Boltzmann equations for the massive neutrinos are discussed in e.g. [23].

It is assumed that the CDM particles are weakly interacting. Therefore they are described by a collisionless Boltzmann equation. It can be derived in a similar fashion as the collisionless Boltzmann equation for the photons, except that the large non- zero rest mass of the CDM particles needs to be accounted for. Taking the 0^th and

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1^st moments of the Boltzmann equation (operating with R

d³p and R

d³pˆpⁱ on both sides of the Boltzmann equation) we obtain the equations for the density contrast and the velocity perturbation of the CDM. Because p/E of the CDM particles is small (due to large rest mass), the higher order multipoles (`≥2) are insignificant.

The collisionless part of the baryon Boltzmann equation is identical to the Boltz- mann equation of CDM. The baryon interactions with other baryons and photons contribute to the collision term.

The Boltzmann equations for the massless neutrinos (subscript ”ν”), CDM (subscript ”c”) and baryons (subscript ”b”) are

δ_ν⁰ + 4

3kvν −4Ψ⁰ = 0 (2.64)

v⁰_ν+ k

6Πν − k

4δν −kΦ = 0 (2.65)

δ_c⁰ +kvc−3Ψ⁰ = 0 (2.66)

v_c⁰ +vc−kΦ = 0 (2.67)

δ⁰_b+kvb−3Ψ⁰ = 0 (2.68)

v_b⁰ +vb−kΦ = −τ⁰4ρ_γ 3ρb

(vγ−vb). (2.69)

There is a similar hierarchy for neutrinos as for photons, except for the collision term.

2.5 C

_`

Spectrum

The Boltzmann hierarchy of CMB photons (Eqs. (2.60) - (2.63)), Boltzmann equations for neutrinos, CDM and baryons (Eqs. (2.64) - (2.69)) and Einstein equations for Ψ and Φ (Eqs. (2.15) - (2.18)) constitute a complete set of equations, where the multipoles of the photon brightness function can, in principle, be solved for any time η after the photon decoupling. Due to the large number of intercoupled equations this is, in general, a tedious task. In Sect. 2.6 we will discuss the line-of-sight in- tegration ([26]), which is an approach that makes this problem more tractable and enables more efficient numerical computations.

For the time being let us assume that we have been able to find the values that the multipoles Θ_`(η,k) have today (at η = η₀). The CMB temperature anisotropy δT(η0,x= 0,p), that we detect today, depends on the photon brightness function:ˆ δT(η0,x= 0,p) =ˆ T0Θ(η0,x= 0,p). Hereˆ T0 is the background CMB temperature (T₀ = 2.725 K) and x= 0 is the location of the observer.

The CMB temperature anisotropy can be expressed in terms of the Fourier mode expansion of the photon brightness function (cf. Eq. (2.12))

δT(η0,p) =ˆ δT(η0,x= 0,p) =ˆ T0Θ(η0,x= 0,p) =ˆ T0

(2π)^3/2 Z

d³kΘ(η0,k, µ).

(2.70)

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We can insert the multipole expansion (Eq. (2.54)) in place of Θ(η0,k, µ). Before doing that we use the identity

(2`+ 1)P`(µ) = (2`+ 1)P`(ˆk·p) = 4πˆ X` m=−`

Y_`m^∗ (ˆk)Y`m(ˆp) (2.71)

and substitute (2` + 1)P`(µ) with 4πP`

m=−`Y_`m^∗ (ˆk)Y`m(ˆp) in the expansion. We obtain now for the anisotropy

δT(η0,p) =ˆ X

`m

· 4πT₀ (2π)^3/2(−i)^`

Z

d³kΘ`(η0,k)Y_`m^∗ (ˆk)

¸

Y`m(ˆp). (2.72) Here the `-sum goes from 0 to ∞ and the m-sum from −` to ` (at each `). The CMB anisotropy δT(η0,p) is a function on the celestial sphere and we have aboveˆ its expansion in terms of the spherical harmonics Y`m(ˆp). We can extract the a`m

expansion coefficients from Eq. (2.72) a`m = 4πT0

(2π)^3/2(−i)^` Z

d³kΘ`(η0,k)Y_`m^∗ (ˆk). (2.73) Because we assume first order perturbation theory, the coefficients Θ_`(η₀,k) depend linearly on the primordial curvature perturbations (see Sect. 2.2)

Θ`(η0,k)≡T`(η0, k)R^k(rad). (2.74) This equation defines the transfer function T`(η0, k). It can be calculated using the equations we have discussed above. Note that it depends on the magnitude of the wavevector and the direction dependence is in the primordial perturbations only.

Because we assumed that the primordial curvature perturbations are complex zero mean Gaussian distributed random variables, the a`m are complex, zero mean and Gaussian distributed as well.

Under the assumption of a Gaussian distribution the covariance ha`ma^∗_`0m⁰i contains the full statistical description of the CMB temperature anisotropy. The covariance is

ha_`ma^∗_`0m⁰i= 2T₀²

π (−i)^`(i)^`⁰ Z

d³kd³k⁰Y_`m^∗ (ˆk)Y_`⁰_m⁰( ˆk⁰)hΘ_`(η₀,k)Θ^∗_`0(η₀,k⁰)i, (2.75) where the expectation value hΘ`Θ^∗_`0i can be expressed in terms of the primordial power spectrum PR(k) (see Eq. (2.23))

hΘ`(η0,k)Θ^∗_`0(η0,k⁰)i=T`(η0, k)T_`^∗0(η0, k⁰)hR^k(rad)R^∗k⁰(rad)i=

= 2π²

k³ T`(η0, k)T_`^∗0(η0, k⁰)P^R(k)δ(k−k⁰). (2.76)

Map-Making and Power Spectrum Estimation for Cosmic Microwave Background Temperature Anisotropies